H. L. Royden, see [1]}, showed that the action (REF ) of \(\Gamma _g\)  on \(\mathcal {T}_g\)  is by biholomorphic automorphisms and the group of biholomorphic automorphims of \(\mathcal {T}_g\) , for genus \(g\geqslant 3\) , can be identified with \(\Gamma _g.\)  The group of biholomorphic automorphims of \(\mathcal {T}_2\)  is equivalent to \(\Gamma _2\)  modulo the hyperelliptic involution. Moreover, the action of \(\Gamma _g\)  on \(\mathcal {T}_g\)  is properly discontinuous but not free, see theorem 6.18 in [2]}, or proposition 1.13 in [3]} for the proof. The quotient
\(\mathcal {M}_g := \mathcal {T}_g / \Gamma _g\) 
